New results for the SQCD Hilbert series

Jokela, Niko; Järvinen, Matti; Keski-Vakkuri, Esko

doi:10.1007/jhep03(2012)048

Cited by 7 publications

(20 citation statements)

References 56 publications

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“…This paper is divided into four main parts. The first part, containing Section 2, develops the connection between Hilbert series and the partition function of the Coulomb gas in two dimensions, which was established for the U (N c ) and SU (N c ) gauge groups in [12]. The basic connection goes back to the seminal works of Wigner and Dyson [20][21][22][23][24][25].…”

Section: Outline and Key Resultsmentioning

confidence: 99%

“…By examining the behavior of the free energy at the transition point between these two phases, we demonstrate the possibility of the third order phase transition, reminiscient of [18,19]. This phase transition was uncovered for the U (N c ) and SU (N c ) gauge groups in [12].…”

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confidence: 84%

“…These expressions are explicitly written in (4.9) of [6], (3.14) of [7], (3.6) of [10], (2.8), (3.3), (4.4) of [11], and (5), (8) of [12]. In [10], [11], and [12], it was shown that these multi-contour integrals can be recast in terms of determinants of Toeplitz matrices for unitary and special unitary gauge groups, and determinants of Hankel matrices for the special orthogonal and symplectic gauge groups. Writing these integrals in terms of determinants has several advantages: it allows for exact expressions for the Hilbert series for arbitrary number of colours and flavours, as well as various asymptotic results when the number of colours and flavours are large.…”

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confidence: 99%

“…• If N f /N c is fixed in a certain interval, known as the conformal window, the theory has a non-trivial infrared fixed point. For SU (N c ), Sp(N c ), and SO(N c ) gauge groups, the conformal windows are 3 2 N c < N f < 3N c , 3 2 (N c + 1) < N f < 3(N c + 1), and The moduli space of U (N c ) and SU (N c ) SQCD with N f flavours in this asymptotic limit was studied in detail in [10,12]. In this paper, we focus on SQCD with SO(N c ) and Sp(N c ) gauge groups.…”

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confidence: 99%

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Moduli space of supersymmetric QCD in the Veneziano limit

Chen

Jokela

Järvinen

et al. 2013

J. High Energ. Phys.

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We study the moduli space of 4d N=1 supersymmetric QCD in the Veneziano limit using Hilbert series. In this limit, the numbers of colours and flavours are taken to be large with their ratio fixed. It is shown that the Hilbert series, which is a partition function of an ensemble of gauge invariant quantities parametrising the moduli space, can also be realised as a partition function of a system of interacting Coulomb gas in two dimensions. In the electrostatic equilibrium, exact and asymptotic analyses reveal that such a system exhibits two possible phases. Physical quantities, such as charge densities, free energies, and Hilbert series, associated with each phase, are computed explicitly and discussed in detail. We then demonstrate the existence of the third order phase transition in this system.Comment: 38 pages; v2: clarifications added, published versio

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Section: Outline and Key Resultsmentioning

confidence: 99%

mentioning

confidence: 84%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Moduli space of supersymmetric QCD in the Veneziano limit

Chen

Jokela

Järvinen

et al. 2013

J. High Energ. Phys.

Self Cite

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“…A host of activity ensued, developing and refining various aspects of the programme [22][23][24][25][26][27][28][29][30][31][32][33][34], even using the Hilbert-series technology to (the regular, non-supersymmetric) Standard-Model phenomenology [35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Standard model plethystics

Xiao

Matti

2019

Phys. Rev. D

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We study the vacuum geometry prescribed by the gauge invariant operators of the MSSM via the Plethystic Programme. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the MSSM gauge invariants. All data in Mathematica format are also presented. * Yan.Xiao@city.ac.uk † hey@maths.ox.ac.uk

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Quivers, words and fundamentals

Mattioli

Ramgoolam

2015

J. High Energ. Phys.

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A systematic study of holomorphic gauge invariant operators in general N = 1 quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.

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New results for the SQCD Hilbert series

Cited by 7 publications

References 56 publications

Moduli space of supersymmetric QCD in the Veneziano limit

Moduli space of supersymmetric QCD in the Veneziano limit

Standard model plethystics

Quivers, words and fundamentals

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